reserve n,m,i,k for Element of NAT;
reserve x,X,X1 for set;
reserve r,p for Real;
reserve s,x0,x1,x2 for Real;
reserve f,f1,f2 for PartFunc of REAL,REAL n;
reserve h for PartFunc of REAL,REAL-NS n;
reserve W for non empty set;

theorem
  f is total & (for x1,x2 holds f/.(x1+x2) = f/.x1 + f/.x2) &
  (ex x0 st f is_continuous_in x0) implies f|REAL is continuous
proof
  assume A1:
  f is total & (for x1,x2 holds f/.(x1+x2) = f/.x1 + f/.x2);
  given x0 such that
A2: f is_continuous_in x0;
  reconsider g= f as PartFunc of REAL,REAL-NS n
  by REAL_NS1:def 4;
A3: now let x1,x2;
A4:  g/.x1 = f/.x1 & g/.x2 = f/.x2 by REAL_NS1:def 4;
      thus g/.(x1+x2) = f/.(x1+x2) by REAL_NS1:def 4
      .=f/.x1 + f/.x2 by A1
      .= g/.x1 + g/.x2 by A4,REAL_NS1:2;
    end;
    g is_continuous_in x0 by A2;
    then g|REAL is continuous by A1,A3,NFCONT_3:23;
    hence thesis by Th23;
  end;
